Some Open Problems in the Symmetry of Graphs

When dealing with symmetry properties of combinatorial objects, such as vertex-transitive (di)graphs, a fundamental question is to determine their full automorphism group. Many of these objects naturally display certain inherently obvious symmetries. It is often the case, however, that additional symmetries, though hidden or difficult to grasp, are present.

When this is the case, the goal is to find a reason for their existence and a method for their description.

A simple yet non/trivial question along these lines concerns the dichotomy of even/odd automorphisms, wherean automorphism of a graph $X$ is said to be odd (resp. even) if it acts as an odd (resp. even) permutation on the vertex set of $X$.

Deciding in general whether a given (vertex-transitive) graph has odd automorphisms or not is indeed a non-trivial problem.

For example, one of the consequences of the classification of finite simple groups (CFSG) is that $A_5$ and $S_5$ are the only simply primitive groups of degree twice a prime number. Therefore, the Petersen graph and its complement are the only examples of connected vertex-transitive graphs of order twice a prime number with a simply primitive automorphism group. Consequently, every such graph has an odd automorphism. I am aware of no CFSG-free proof of this result to exist.

This brings us to the second goal of this talk: we will discuss possibile ways of replacing exisisting proofs of certain results in algebraic graph theory, that rely on CFSG, with direct arguments.

Most of the results discussed in this talk are joint with my colleagues from the University of Primorska: Ademir Hujdurović, Klavdija Kutnar and Stefko Miklavič.